We propose novel compression algorithms for time-varying channel state information (CSI) in wireless communications. The proposed scheme combines (lossy) vector quantisation and (lossless) compression. First, the new vector quantisation technique is based on a class of parametrised companders applied on each component of the normalised CSI vector. Our algorithm chooses a suitable compander in an intuitively simple way whenever empirical data are available. Then, the sequences of quantisation indices are compressed using a context-tree-based approach. Essentially, we update the estimate of the conditional distribution of the source at each instant and encode the current symbol with the estimated distribution. The algorithms have low complexity, are linear-time in both the spatial dimension and time duration, and can be implemented in an online fashion. We run simulations to demonstrate the effectiveness of the proposed algorithms in such scenarios.